Scott analysis, linear orders and almost periodic functions

TL;DR

This paper constructs linear orders with specific Scott complexities, completing the classification and providing new examples, including rigid structures, with a focus on the complexity levels of their descriptions.

Contribution

It introduces a method to construct linear orders with Scott complexity a_{+1} for any limit ordinal
b, filling gaps in the classification and producing new examples of high complexity structures.

Findings

Constructed linear orders with Scott complexity a_{+1} for all limit
b

Produced continuum-many structures a_{+1} that are
b-equivalent

Demonstrated non-existence of structures with certain Scott complexities at specific levels

Abstract

For any limit ordinal $\lambda$, we construct a linear order $L_\lambda$ whose Scott complexity is $\Sigma_{\lambda+1}$. This completes the classification of the possible Scott sentence complexities of linear orderings. Previously, there was only one known construction of any structure (of any signature) with Scott complexity $\Sigma_{\lambda+1}$, and our construction gives new examples, e.g., rigid structures, of this complexity. Moreover, we can construct the linear orders $L_\lambda$ so that not only does $L_\lambda$ have Scott complexity $\Sigma_{\lambda+1}$, but there are continuum-many structures $M \equiv_\lambda L_\lambda$ and all such structures also have Scott complexity $\Sigma_{\lambda+1}$. In contrast, we demonstrate that there is no structure (of any signature) with Scott complexity $\Pi_{\lambda+1}$ that is only $\lambda$-equivalent to structures with Scott complexity $\Pi_{\lambda+1}$. Our construction is based on functions $f \colon \mathbb{Z}\to \mathbb{N}\cup \{\infty\}$ which are almost periodic but not periodic, such as those arising from shifts of the $p$-adic valuations.